2016 AMC 10B Problem 14

A square must lie above $y=-0.1$, to the left of $x=5.1$, and below $y=\pi x$. Since $\pi$ is a little more than $3$, the lattice heights available at $x=1,2,3,4$ are $3,6,9,12$, and side lengths larger than $3$ cannot fit. Count by side length using the top-left lattice point. For side length $1$, there are $3+6+9+12=30$ choices. For side length $2$, there are $2+5+8=15$ choices. For side length $3$, there are $1+4=5$ choices. The total is $30+15+5=50$. Thus, the correct answer is D.